Expanding a square

Average Error: 39.4 → 0.0

Time: 1.3s

Precision: binary64

Cost: 448

Math

\[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

↓

\[x \cdot 2 + x \cdot x \]

FPCore

(FPCore (x) :precision binary64 (- (* (+ x 1.0) (+ x 1.0)) 1.0))

↓

(FPCore (x) :precision binary64 (+ (* x 2.0) (* x x)))

C

double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

↓

double code(double x) {
	return (x * 2.0) + (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) * (x + 1.0d0)) - 1.0d0
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 2.0d0) + (x * x)
end function

Java

public static double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

↓

public static double code(double x) {
	return (x * 2.0) + (x * x);
}

Python

def code(x):
	return ((x + 1.0) * (x + 1.0)) - 1.0

↓

def code(x):
	return (x * 2.0) + (x * x)

Julia

function code(x)
	return Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - 1.0)
end

↓

function code(x)
	return Float64(Float64(x * 2.0) + Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) * (x + 1.0)) - 1.0;
end

↓

function tmp = code(x)
	tmp = (x * 2.0) + (x * x);
end

Wolfram

code[x_] := N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[x_] := N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.4

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

2. Simplified 0.0

   \[\leadsto \color{blue}{x \cdot \left(x + 2\right)} \]

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{x \cdot 2 + x \cdot x} \]

4. Final simplification 0.0

   \[\leadsto x \cdot 2 + x \cdot x \]

Alternatives

Alternative 1
Error	1.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3640.325933075684:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 0.0029421815227809534:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

\[x \cdot \left(x + 2\right) \]

Alternative 3
Error	21.0
Cost	192

\[x \cdot 2 \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))